Random matrices and the average topology of the intersection of two quadrics

TL;DR

This paper investigates the topological complexity of intersections of random quadrics in real projective space, showing that their Betti numbers grow linearly with dimension, using a blend of random matrix theory and algebraic topology.

Contribution

It establishes the asymptotic behavior of Betti numbers for intersections of two random quadrics, combining techniques from random matrix theory and algebraic geometry.

Findings

Betti numbers grow linearly with dimension n

Asymptotic behavior of topological invariants for random quadrics

Methodology combines random matrix theory, integral geometry, and spectral sequences

Abstract

Let X_R be the zero locus in RP^n of one or two independently and Weyl distributed random real quadratic forms (this is the same as requiring that the corresponding symmetric matrices are in the Gaussian Orthogonal Ensemble). We prove that the sum of the Betti numbers of X_R behaves asymptotically as n (when n goes to infinity). The methods we use combine Random Matrix Theory, Integral geometry and spectral sequences.